====== Relaxed Assumptions ======

\( \require{AMSsymbols} \)
\( \require{newcommand} \)
\( \newcommand{\abs}[1]{\left|#1\right|} \)
\( \newcommand{\diff}[2]{\frac{d{#1}}{d{#2}}} \)

====== Overview ======

In this section I am again trying to the next "higher order correction" to the logistic equation.  I reason that the only assumptions I need are:
  * interactions are at most pairwise, and
  * physiological processes (changes of state, not population size) are much more frequent than ecological (changes of population size).

I am able to derive a complex equation for the dynamics of the whole population (independent of state).  Currently, I'm working on simplifying the rate equation, perhaps by adding more assumptions.  I expect the techniques derived in [[research:draft:general_allee:start|]] will be useful.

====== Background ======

The gist of my argument (eg. in [[research:draft:2011:implicit_competition:start|]]) so far goes like this:

  * the logistic equation is usually descriptive
  * but it can be derived from first principles, using the following assumptions:
    - max pairwise interactions
    - only one individual affected by interaction (eg. birth or death)
    - one type
  * this very coarse approximation yields logistic behaviour
  * how can we extend this model?
  * expect higher than pairwise interactions rare so don't change that
  * //allowing more than one individual to be affected leads to combinatorial explosion of reactions, so not a good choice//
  * hence, left with increasing the number of types

The problem is that I can't think of a good "biological" reason not to allow more than one individual to be affected in an reaction.  My rationale is purely for tractability -- but nature doesn't really care about what I can solve mathematically.  If it happens, it happens.  So, I should relax this assumption and see what happens (and how far I can get in my analysis).  

Another reason I was trying to keep things simple was to help make the model empirically-testable: if I could establish a simple mapping from rate constants to population-level parameters (eg. \(r\; \&\; K\)) then I could argue that field biologists could -- at least in principle -- measure the rate constants and compare them against expectation.  But I found in [[research:draft:2011:implicit_competition:start]] that it wasn't feasible.  The rate constants mapped onto some other parameters (\(u_0, u_1,\) etc.) in a complicated way and those parameters had a nontrivial mapping onto measurable population-level quantities.  So the relationship was muddied and it wasn't clear it would work -- again, there wasn't a strong enough motive to keep this strong assumption.

====== Assumptions ======

The following assumption will hold below:

<WRAP center round important 60%>
Reactions are spontaneous (one reactant) or pairwise (two reactants), no higher order.
</WRAP>

We will look at the cases where there are one or two types.

====== One type ======

With the above assumption and only one type of individual any of the following reactions are possible:
\[
\begin{array}{rcll}
  N   & \xrightarrow{c^{(n)}_i}    & i N     & i=0,1,2,\ldots  \\
  2 N & \xrightarrow{c^{(nn)}_j}     & j N     & j=0,1,2,\ldots
\end{array}
\]

From the law of mass action the dynamics follow
\[
  \diff{n}{t} = \sum_{i=0}^\infty c^{(n)}_i (i-1) n + \sum_{j=0}^\infty c^{(nn)}_j (j-2) n^2
\]
which can be expressed as
\[
  \diff{n}{t} = r n (1 - n/K)
\]
where
\[
\begin{array}{rcl}
  r & = & \sum_{i>1} c^{(n)}_i (i-1) - c^{(n)}_0 \\
  K & = & \frac{r}{2 c^{(nn)}_0 + c^{(nn)}_1 - \sum_{j>2} c^{(nn)}_j (j-2)}
\end{array}
\]
Hence, the relaxed assumptions //still// give the logistic equation.  If \(r,K>0\) then the population sustains with a carrying capacity \(K\).

Also note we could have written the reactions as
\[
\begin{array}{rcl}
  N   & \xrightarrow{\alpha}    & 2 N \\
  N   & \xrightarrow{\delta}    & \emptyset \\
  2 N & \xrightarrow{\beta}     & 3 N \\
  2 N & \xrightarrow{\chi}      & N
\end{array}
\]
where
\[
\begin{array}{rcl}
  \alpha & = & \sum_{i>1} c^{(n)}_i (i-1) \\
  \beta  & = & \sum_{j>2} c^{(nn)}_j (j-2) \\
  \delta & = & c^{(n)}_0 \\
  \chi   & = & 2 c^{(nn)}_0 + c^{(nn)}_1
\end{array}
\]
without any loss of generality.

====== Two types ======

Using the same assumptions but with two types of individuals the possible reactions expand to
\[
\begin{array}{rclr}
  A & \xrightarrow{c^{(a)}_{ij}}    & i A + j B       & i,j=0,1,2,\ldots  \\
  B & \xrightarrow{c^{(b)}_{kl}}    & k A + l B       & k,l=0,1,2,\ldots  \\
  2 A   & \xrightarrow{c^{(aa)}_{mn}}    & m A + n B  & m,n=0,1,2,\ldots  \\
  A + B & \xrightarrow{c^{(ab)}_{pq}}    & p A + q B  & p,q=0,1,2,\ldots  \\
  2 B   & \xrightarrow{c^{(bb)}_{rs}}    & r A + s B  & r,s=0,1,2,\ldots
\end{array}
\]

The dynamics become
\[
\begin{array}{rcl}
  \diff{a}{t} & = & \sum_{i,j} c^{(a)}_{ij} (i-1) a + \sum_{k,l} c^{(b)}_{kl} k b + \sum_{m,n} c^{(aa)}_{mn} (m-2) a^2 \\
  & & + \sum_{p,q} c^{(ab)}_{pq} (p-1) a b + \sum_{r,s} c^{(bb)}_{rs} r b^2 \\
  \diff{b}{t} & = & \sum_{i,j} c^{(a)}_{ij} j a + \sum_{k,l} c^{(b)}_{kl} (l-1) b + \sum_{m,n} c^{(aa)}_{mn} n a^2 \\
  & & + \sum_{p,q} c^{(ab)}_{pq} (q-1) a b + \sum_{r,s} c^{(bb)}_{rs} (s-2) b^2.
\end{array}
\]

These dynamics can also be written as
\[
\begin{array}{rcl}
  \diff{a}{t} & = & \alpha_{aa} a - \delta_a a + \alpha_{ba} b + \beta_{aaa} a^2 - \chi_{aa} a^2 + \beta_{aba} a b - \chi_{ab} a b + \beta_{bba} b^2 \\
  \diff{b}{t} & = & \alpha_{ab} a + \alpha_{bb} b - \delta_b b + \beta_{aab} a^2 + \beta_{abb} a b - \chi_{ba} a b + \beta_{bbb} b^2 - \chi_{bb} b^2
\end{array}
\]
where
\[
\begin{array}{rclrcl}
  \alpha_{aa} & = & \sum_{i>1,j} c^{(a)}_{ij} (i-1)  &  \alpha_{ab} & = & \sum_{i,j} c^{(a)}_{ij} j \\
  \alpha_{ba} & = & \sum_{k,l} c^{(b)}_{kl} k        &  \alpha_{bb} & = & \sum_{k,l>1} c^{(b)}_{kl} (l-1) \\
  \beta_{aaa} & = & \sum_{m>2,n} c^{(aa)}_{mn} (m-2) &  \beta_{aab} & = & \sum_{m,n} c^{(aa)}_{mn} n \\
  \beta_{aba} & = & \sum_{p>1,q} c^{(ab)}_{pq} (p-1) &  \beta_{abb} & = & \sum_{p,q>1} c^{(ab)}_{pq} (q-1) \\
  \beta_{bba} & = & \sum_{r,s} c^{(bb)}_{rs} r       &  \beta_{bbb} & = & \sum_{r,s>2} c^{(bb)}_{rs} (s-2) \\
  \delta_a    & = & \sum_j c^{(a)}_{0j}              &  \delta_b    & = & \sum_k c^{(b)}_{k0} \\
  \chi_{aa}   & = & \sum_n (2 c^{(aa)}_{0n} + c^{(aa)}_{1n}) & \chi_{bb} & = & \sum_r (2 c^{(bb)}_{r0} + c^{(bb)}_{r1}) \\
  \chi_{ab}   & = & \sum_q c^{(ab)}_{0q}             &  \chi_{ba}   & = & \sum_p c^{(ab)}_{p0}.
\end{array}
\]

So, without loss of generality, we could have captured all possible dynamics with just the following 16 reactions:
\[
\begin{array}{rclrcl}
  A   & \xrightarrow{\alpha_{aa}}    & 2 A    &  A   & \xrightarrow{\alpha_{ab}}    & A + B \\
  B   & \xrightarrow{\alpha_{ba}}    & A + B  &  B   & \xrightarrow{\alpha_{bb}}    & 2 B \\
  2 A & \xrightarrow{\beta_{aaa}}    & 3 A    &  2 A & \xrightarrow{\beta_{aab}}    & 2 A + B \\
  A+B & \xrightarrow{\beta_{aba}}    & 2A + B &  A+B & \xrightarrow{\beta_{abb}}    & A + 2 B \\
  2 B & \xrightarrow{\beta_{bba}}    & A + 2B &  2 B & \xrightarrow{\beta_{bbb}}    & 3 B \\
  A   & \xrightarrow{\delta_a}    & \emptyset &  B   & \xrightarrow{\delta_b}    & \emptyset \\
  2 A & \xrightarrow{\chi_{aa}}      & A      &  2 B & \xrightarrow{\chi_{bb}}      & B \\
  A+B & \xrightarrow{\chi_{ab}}      & B      &  A+B & \xrightarrow{\chi_{ba}}      & A.
\end{array}
\]

====== Transitions ======

Each of the 16 reactions above represents a birth or death process.  Other common processes we should consider (but were lost in our simplification) are transitions from one type to another.  Let's add those back in:
\[
\begin{array}{rclrcl}
  A   & \xrightarrow{\phi_a}      & B      &  B   & \xrightarrow{\phi_b}      & A \\
  2 A & \xrightarrow{\nu_{aa}}    & A + B  &  2 B & \xrightarrow{\nu_{bb}}    & A + B \\
  A+B & \xrightarrow{\nu_{ab}}    & 2 B    &  A+B & \xrightarrow{\nu_{ba}}    & 2 A.
\end{array}
\]

Then
\[
\begin{array}{rcl}
  \diff{a}{t} & = & \alpha_{aa} a - \delta_a a + \alpha_{ba} b + \beta_{aaa} a^2 - \chi_{aa} a^2 + \beta_{aba} a b - \chi_{ab} a b + \beta_{bba} b^2 \\
              &   & - \phi_a a + \phi_b b - \nu_{aa} a^2 + \nu_{bb} b^2 - \nu_{ab} a b + \nu_{ba} a b \\
  \diff{b}{t} & = & \alpha_{ab} a + \alpha_{bb} b - \delta_b b + \beta_{aab} a^2 + \beta_{abb} a b - \chi_{ba} a b + \beta_{bbb} b^2 - \chi_{bb} b^2 \\
              &   & + \phi_a a - \phi_b b + \nu_{aa} a^2 - \nu_{bb} b^2 + \nu_{ab} a b - \nu_{ba} a b.
\end{array}
\]
Note that the definitions of \(\alpha, \beta, \delta, \& \chi\) given above no longer apply -- they would need to be modified.

The motivation for applying these transition processes is that they don't change the population density, \(n=a+b\), so we can separate out the dynamics for the density \(n\) and the frequency \(f=a/n\):
\[
\begin{array}{rcl}
  \diff{n}{t} & = & \diff{a}{t} + \diff{b}{t} \\
  \diff{f}{t} & = & \frac{1}{n} \diff{a}{t} - \frac{f}{n} \diff{n}{t}.
\end{array}
\]
With substitution and some algebra we find
\[
\begin{array}{rl}
  \diff{n}{t} = & n \left[ f (A_a + \alpha_{ab}) + (1-f) (\alpha_{ba} + A_b) \right] \\
                & - n^2 \left[ f^2 (X_{aa} - \beta_{aab}) + f (1-f) (X_{ab} + X_{ba}) + (1-f)^2 (X_{bb} - \beta_{bba}) \right] \\
  \diff{f}{t} = & -f \phi_{a} + (1-f) \phi_{b} - f^2(\alpha_{ab}+n \nu_{aa}) + f(1-f)(A_a-A_b - n N) \\ 
                & + (1-f)^2 (\alpha_{ba} + n \nu_{bb}) - f^3 n \beta_{aab} + f^2 (1-f) n (X_{ba} - X_{aa}) \\
                & + f(1-f)^2n (X_{bb} - X_{ab})+(1-f)^3 n \beta_{bba}
\end{array}
\]
where we have defined
\[
\begin{array}{rcl}
  A_a    & = & \alpha_{aa} - \delta_a \\
  A_b    & = & \alpha_{bb} - \delta_b \\
  N      & = & \nu_{ab} - \nu_{ba} \\
  X_{aa} & = & \chi_{aa} - \beta_{aaa} \\
  X_{ab} & = & \chi_{ab} - \beta_{aba} \\
  X_{ba} & = & \chi_{ba} - \beta_{abb} \\
  X_{bb} & = & \chi_{bb} - \beta_{bbb}.
\end{array}
\]

For some combinations of rate constants we will find that \(n\) decouples from \(f\).  Namely, if all of the following hold,
\[
\begin{array}{rcl}
  A_a + \alpha_{ab} & = & \alpha_{ba} + A_b \\
  X_{ab} + X_{ba} & = & 2 (X_{bb} - \beta_{bba}) \\
  X_{aa} - \beta_{aab} & = & X_{bb} - \beta_{bba},
\end{array}
\]
	
then \(\diff{n}{t} = n (\alpha_{ba} + A_b) - n^2 ( X_{bb} - \beta_{bba} )\), which is independent of \(f\) and is just a logistic equation.  The purpose of our investigation is to explore behaviour beyond logistic that emerges from resolving two types (logistic behaviour is already observed for one type) so we will restrict ourselves to the case where
<WRAP center round important 60%>
At least one of the following holds:
\[
\begin{array}{rcl}
  \alpha_{aa} + \alpha_{ab} - \delta_a & \neq & \alpha_{ba} + \alpha_{bb} - \delta_b \\
  & \text{or} & \\
  \chi_{ab} - \beta_{aba} + \chi_{ba} - \beta_{abb} & \neq & 2 (\chi_{bb} - \beta_{bba} - \beta_{bbb}) \\
  & \text{or} & \\
  \chi_{aa} - \beta_{aaa} - \beta_{aab} & \neq & \chi_{bb} - \beta_{bba} - \beta_{bbb}. &
\end{array}
\]
</WRAP>
	
====== Separation of timescales ======

The full dynamics may be difficult to characterize. But it is more manageable if we assume that the dynamical variables, \(n\) and \(f\), change at drastically different speeds. Let's consider the case where the “ecological” processes are very slow.  We assume:
<WRAP center round important 60%>
"Physiological" processes (where individuals change their character, or type) are much more frequent than "ecological" (where the population density changes).
</WRAP>

Then we can define new "slow parameters" -- denoted by tildes -- as follows:
\[
\begin{array}{rcl}
  \alpha & = & \epsilon \tilde{\alpha} \\
  \beta  & = & \epsilon \tilde{\beta} \\
  A      & = & \epsilon \tilde{A} \\
  X      & = & \epsilon \tilde{X}
\end{array}
\]
where \(\epsilon \ll 1\).  More specifically, we choose \(\epsilon\) small enough so that every term in the rate equations with an \(\epsilon\) is insignificant compared to those without.

On the slow timescale, \(\tilde{t} = \epsilon t\), we have (where \(x' = \diff{x}{\tilde{t}} = \frac{1}{\epsilon}\diff{x}{t}\))
\[
\begin{array}{rl}
  n' = & n \left[ f (\tilde{A}_a + \tilde{\alpha}_{ab}) + (1-f) (\tilde{A}_b + \tilde{\alpha}_{ba}) \right] \\
       & - n^2 \left[ f^2 (\tilde{X}_{aa} - \tilde{\beta}_{aab}) + f (1-f) (\tilde{X}_{ab}  + \tilde{X}_{ba}) + (1-f)^2 (\tilde{X}_{bb} - \tilde{\beta}_{bba}) \right] \\
  f' = & \frac{1}{\epsilon} \left[ - f \phi_a + (1-f) \phi_b - n \left( f^2 \nu_{aa} + f (1-f) N - (1-f)^2 \nu_{bb} \right) \right] + O[\epsilon^0].
\end{array}
\]

Notice that \(n\) varies only on the slow timescale but \(f\) varies quickly (because of the \(\epsilon^{-1}\) term).  To lowest order((We can explore higher order corrections later.)) in \(\epsilon\) on the fast timescale
\[
\begin{array}{rcl}
  \diff{f}{t} & \approx & - f \phi_a + (1-f) \phi_b - n \left( f^2 \nu_{aa} + f (1-f) N - (1-f)^2 \nu_{bb} \right) \\
      & \approx & f^2 n (\nu_{bb} + N - \nu_{aa}) - f (2 n \nu_{bb} + n N + \phi_a + \phi_b) + n \nu_{bb} + \phi_b \\
      & \approx & a_f f^2 + b_f f + c_f
\end{array}
\]
where
\[
\begin{array}{rcl}
  a_f & = & n (\nu_{bb} + N - \nu_{aa}) \\
  b_f & = & - (2 n \nu_{bb} + n N + \phi_a + \phi_b) \\
  c_f & = & n \nu_{bb} + \phi_b
\end{array}
\]

====== Fast dynamics ======

On the fast timescale \(n\) is effectively constant so we can study the dynamics of \(f\) alone.  How \(f\) behaves depends on the parameters \(a_f, b_f, \;\&\; c_f\).  First we notice that \(c_f\geq 0\), reducing the possible dynamics.  Next, let us define
\[
  d_f \equiv n \nu_{aa} + \phi_a \geq 0.
\]
Then
\[
  b_f = -a_f - c_f - d_f
\]
so \(a_f + b_f\leq 0\).

===== Degenerate case: Linear =====

Before we explore the properties of the full quadratic rate equation for \(f\) we should consider the degenerate case where the quadratic term is null, \(a_f=0\), so the rate equation is linear:
\[
  \diff{f}{t} = b_f f + c_f.
\]
In this case, \(b_f = -c_f - d_f\) so there is a single stable((The equilibrium is stable because \(b_f\leq 0\).)) equilibrium at
<WRAP center round info 60%>
\[
  f^* = \frac{c_f}{c_f + d_f} = \frac{n \nu_{bb} + \phi_b}{n (\nu_{aa} + \nu_{bb}) + \phi_a + \phi_b}
\]
if \(a_f=0\) (\(\nu_{aa} + \nu_{ba}=\nu_{ab} + \nu_{bb}\)).
</WRAP>
which always occurs in the range \(0\leq f^* \leq 1\).

===== Generic case: Quadratic =====

On the other hand, if \(a_f \neq 0\), the equilibria for the fast dynamics are given by \(\diff{f}{t}=0\), or
\[
  f_\pm = \frac{-b_f \pm \sqrt{R}}{2 a_f}
\]
where
\[
\begin{array}{rcl}
  R & \equiv & b_f^2 - 4 a_f c_f \\
    & = & a_f^2 + c_f^2 + d_f^2 + 2 a_f d_f + 2 c_f d_f - 2 a_f c_f \\
    & = & (a_f - c_f + d_f)^2 + 4 c_f d_f \\
    & \geq & 0
\end{array}
\]
is the radicand of \(f_\pm\).  Since \(R\geq 0\) there is always at least one equilibrium, and generally two.  To be (locally) stable (and the expected outcome) we require 
\[
  \left. \diff{}{f} \diff{f}{t} \right|_{f_\pm} = 2 a_f f_\pm + b_f = \pm \sqrt{R} < 0
\]
Conveniently, only the negative root of the radical \(f_-\) is stable so we can neglect the other (unstable) root.  On the fast timescale the dynamics will converge to
\[
  f_- = \frac{-b_f - \sqrt{R}}{2 a_f}.
\]

===== Special case: All B's =====

Considering the limiting case where \(f_- = 0\) we find after some algebra (and excluding the linear case, \(a_f=0\), already considered above) that we require \(c_f=0\) and \(a_f \geq -d_f\).
<WRAP center round info 60%>
\[
\begin{array}{ccc}
  & f_- = 0 \text{ if} & \\
  c_f =0 & \& & a_f \geq -d_f \\
  (\nu_{bb} = \phi_b =0 & \& & n \nu_{ba} \leq n \nu_{ab} + \phi_a).
  
\end{array}
\]
</WRAP>
In this case the only transition producing more \(A\)-types (\(A+B\xrightarrow{\nu_{ba}}2A\)) is dominated by the transitions from \(A\) to \(B\).

===== Special case: All A's =====

Considering the limiting case where \(f_- = 1\) we find after some algebra (and excluding the linear case, \(a_f=0\), already considered above) that we require \(d_f=0\) and \(a_f \leq c_f\).
<WRAP center round info 60%>
\[
\begin{array}{ccc}
  & f_- = 1 \text{ if} & \\
  d_f=0 & \& & a_f\leq c_f \\
  (\nu_{aa}=\phi_a=0 & \& & n \nu_{ab} \leq n \nu_{ba} + \phi_b).
\end{array}
\]
</WRAP>
In this case the only transition producing more \(B\)-types (\(A+B\xrightarrow{\nu_{ab}}2B\)) is dominated by the transitions from \(B\) to \(A\).

===== Special case: Only spontaneous transitions =====

Let's consider the case where we only have spontaneous -- not induced -- reactions, so all \(\nu=0\).  Then we're in the linear regime because \(a_f=0\).  Additionally,
\[
\begin{array}{rcl}
  c_f & = & \phi_b \\
  d_f & = & \phi_a
\end{array}
\]
and we find
\[
  f^* = \frac{\phi_b}{\phi_a + \phi_b}
\]
is a constant.  Therefore, the slow dynamics for \(n\) follow \(n' = r n (1 - n/K)\) for some constants \(r\) and \(K\).  Without induced reactions we recover simple, logistic dynamics.  To observe an Allee effect we require
<WRAP center round important 60%>
\[
  \nu_{aa}+\nu_{bb} > 0. 
\]
</WRAP>


===== Special case: Only induced transitions =====

Similarly, in the case with only induced -- no spontaneous -- reactions, so all \(\phi=0\), we find all the constants \(a_f, b_f, c_f,d_f\propto n\) so all factors of \(n\) cancel out in \(f_-\) leaving a constant value.  As a result, without spontaneous reactions we find just logistic behaviour for \(n\), on the slow timescale.  A mix of both spontaneous and induced transitions are needed to see behaviour beyond logistic:
<WRAP center round important 60%>
\[
  \phi_a + \phi_b > 0.
\]
</WRAP>

===== Two parameters =====

There are many distinct cases to explore.  Which should we focus on first?  As [[wp>William of Ockham]] said: [[http://en.wikiquote.org/wiki/William_of_Occam | “Plurality ought never be posited without necessity.”]]  We might interpret that in terms of preferring explanations with the fewest possible adjustable parameters.  Accurate prediction with fewer adjustable parameters is a more significant result((Less likely to occur by chance.)) than with more.

We need at least one non-zero spontaneous rate constant (one or more of \(\phi_a, \phi_b>0\)) and induced rate constant (one or more of \(\nu_{aa},\nu_{ab},\nu_{ba},\nu_{bb}>0\)) to see deviations from simple logistic behaviour.  At least two parameters (one \(\phi\) and one \(\nu\)) must be "fast".

For simplicity let's reduce the space of parameters to just "fast" or "slow".  A "slow" parameter can be treated as zero since the transition process can then be included in the ecological reactions.  There could be several fast rates but since we want a minimum of parameters we should coarsely group them as being of the same magnitude.  Let us define two parameters, \(\phi>0\) and \(\nu>0\), such that:
  * \(\phi_a,\phi_b = 0\) or \(\phi\), and
  * \(\nu_{aa},\nu_{ab},\nu_{ba},\nu_{bb} = 0\) or \(\nu\).

Then each combination of parameters can yield a different form for the equilibrium frequency, \(f^*\).  We find that the ratio \(\phi / \nu\) sets a natural population density so we shall work in terms of it:
\[
\begin{array}{rcl}
  n^* & \equiv & \phi / \nu \\
  x & \equiv  & n / n^*.
\end{array}
\]
<WRAP right round box 400px>
{{http://sagenb.org/home/Rik_Blok/3/cells/1/fig1.png?400x300}}

All possible combinations of two "fast" parameters results in 12 unique functional dependencies of the quasi-steady state \(A\)-frequency, \(f^*\), on the population density, \(x=n/n^*\).  Figure generated by [[http://sagenb.org/home/Rik_Blok/3/ | this Sage notebook]].
</WRAP>

^  \(\phi_a\)  ^  \(\phi_b\)  ^  \(\nu_{aa}\)  ^  \(\nu_{ab}\)  ^  \(\nu_{ba}\)  ^  \(\nu_{bb}\)  ^  \(f^*(x)\)((Without loss of generality I have only considered cases where \(f^*\) is a decreasing function of \(x\).  The other cases follow from the transform, \(A\leftrightarrow B\).))  ^
|      0       |   \(\phi\)   |       0        |    \(\nu\)     |       0        |       0        | \[ f_1 = \frac{x + 1 - \left| 1 - x \right| }{ 2 x} \]                                                                                                                                       |
|     :::      |     :::      |    \(\nu\)     |       0        |       0        |       0        | \[ f_2 = \frac{ \sqrt{4 x + 1} - 1 }{ 2 x} \]                                                                                                                                                |
|     :::      |     :::      |      :::       |    \(\nu\)     |    \(\nu\)     |      :::       | :::                                                                                                                                                                                          |
|     :::      |     :::      |       0        |       0        |    \(\nu\)     |    \(\nu\)     | \[ f_3 = \frac{3x+1-\left| 1-x \right| }{4x} \]                                                                                                                                              |
|     :::      |     :::      |    \(\nu\)     |       0        |       0        |    \(\nu\)     | \[ f_4 = \frac{x+1}{2 x + 1} \]                                                                                                                                                              |
|     :::      |     :::      |      :::       |    \(\nu\)     |    \(\nu\)     |      :::       | :::                                                                                                                                                                                          |
|     :::      |     :::      |    \(\nu\)     |       0        |    \(\nu\)     |       0        | \[ f_5 = \frac{x - 1 + \sqrt{x^2 + 6x + 1}}{4 x} \]                                                                                                                                          |
|     :::      |     :::      |    \(\nu\)     |    \(\nu\)     |       0        |       0        | \[ f_6 = \frac{1}{x+1} \]                                                                                                                                                                    |
|     :::      |     :::      |    \(\nu\)     |       0        |    \(\nu\)     |    \(\nu\)     | \[ f_7 = - \frac{x - \sqrt{x+1}\sqrt{5x+1}+1}{2 x} \]                                                                                                                                        |
|     :::      |     :::      |    \(\nu\)     |    \(\nu\)     |       0        |    \(\nu\)     | \[ f_8 = \frac{3x + 1 - \sqrt{5x^2+2x+1}}{2x} \]                                                                                                                                             |
|   \(\phi\)   |     :::      |    \(\nu\)     |       0        |       0        |       0        | \[ f_9 = \frac{\sqrt{x+1}-1}{x} \]                                                                                                                                                           |
|     :::      |     :::      |      :::       |    \(\nu\)     |    \(\nu\)     |      :::       | :::                                                                                                                                                                                          |
|     :::      |     :::      |       0        |    \(\nu\)     |       0        |       0        | \[ f_{10} = \frac{x+2-\sqrt{x^2+4}}{2x} \]                                                                                                                                                   |
|     :::      |     :::      |    \(\nu\)     |    \(\nu\)     |       0        |       0        | \[ f_{11} = \frac{1}{x+2} \]                                                                                                                                                                 |
|     :::      |     :::      |    \(\nu\)     |    \(\nu\)     |       0        |    \(\nu\)     | \[ f_{12} = \frac{3x+2-\sqrt{5x^2+8x+4}}{2x} \]                                                                                                                                              |
|                                      all other cases                                       ||||||                                                                                           constant                                                                                           |


The above table can be derived with [[http://sagenb.org/home/Rik_Blok/1/ | this Sage notebook]].

Plotting those 12 cases shows that they are all unique, so there are still too many to derive a general understanding from.  But by plotting all twelve I see the fall into three categories when \(x\ll 1\) and 5 when \(x\gg 1\).  One possible direction of investigation is to solve for 8 limiting cases (3 when \(x\ll 1\) and 5 when \(x\gg 1\)) and then solve slow \(n'\) dynamics for 8 cases.  But I don't think it's a good idea at this time.  The restrictive assumptions I made in this section make it tedious and not generally applicable -- the expected net benefit is small.

A better approach is to throw the most general form of \(f^*\) into the slow dynamics and //then// make some simplifying assumptions (eg. Taylor expansion around \(n=0\)).

====== Slow dynamics ======

Having determined the stable equilibrium for the frequency as a function of a a fixed population density, \(n\), on the fast timescale, \(f(n)=f_-\), we can incorporate this into the dynamics on the slow timescale.

The particular form of \(f(n)\) used will depend on the ecological processes being modeled.  Here are some examples:
  * [[.:linear|Linear]], \(a_f=0\), such as
    * [[.:linear#Estrous cycle]]
    * [[.:linear#Monogamy]]
    * ???
    * [[.:linear#Frequency-independent transitions]], \(\nu_{aa}=\nu_{ab}, \nu_{ba}=\nu_{bb}\), such as
      * [[.:linear#Aggression]] (eg. in lemmings?)
      * ???
  * Quadratic, \(a_f \neq 0\), such as
    * Infection with recovery
  * To be investigated
    * Vulnerable/Safe
